False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

FALSE: every quasi-isometry is continuous, or bijective

Statement refuted

every quasi-isometry is continuous, or bijective.

Facts & Assumptions

Given: The proposed claim together with the witness named in the Statement refuted.

[F1]

A subset is coarsely dense when every point of the space is within a fixed distance of it, and a quasi-isometry is a coarse Lipschitz map admitting a coarse Lipschitz quasi-inverse (Coarsely dense subsets, quasi-inverses and quasi-isometries).

[L1]

A map is $(L, C)$ -coarse Lipschitz when $d (f (x), f (x^{'})) \leq L d (x, x^{'}) + C$ , and an $(L, C)$ -quasi-isometric embedding when in addition $L^{- 1} d (x, x^{'}) - C \leq d (f (x), f (x^{'}))$ (Coarse Lipschitz maps and quasi-isometric embeddings).

[L2]

It is written $⌊ x ⌋$ and called the integer part, or floor, of $x$ . (Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ ).

[L3]

Define $d_{R} : R \times R \to R$ by $d_{R} (x, y) : = ∣ x - y ∣$ . (The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded).

Refutation

technique · contradiction

1.1

The claim asserts that every quasi-isometry is continuous, or that every quasi-isometry is bijective.

F1assume-contra

2.1

The map $f (x) = 2 ⌊ x / 2 ⌋$ is a quasi-isometry of $R$ : the inclusion of the even integers is a coarse Lipschitz quasi-inverse, since $f (2 m) = 2 m$ and every real $x$ satisfies $∣ x - f (x) ∣ < 2$ . But $f$ is not bijective, and it is not continuous at any even integer. So it refutes both readings at once.

F1L1L2L3step 1.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

29 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

C. Loh, Geometric Group Theory: An Introduction (2015 course version), 264 pp. (standard reference, not scraped)
C. Drutu and M. Kapovich, Geometric Group Theory (with an appendix by B. Nica), 837 pp. (standard reference, not scraped)